Which formula defines the conditional probability P(A|B)?

Which formula defines the conditional probability P(A|B)?

• A. P(A|B) = P(A ∩ B) / P(B).
• B. P(A|B) = P(B|A) / P(A).
• C. P(A|B) = P(A) P(B).
• D. P(A|B) = P(A ∪ B) / P(B).

Answer: (A)

Conditional probability tells you the probability of A happening when you already know B has happened. To find that, you focus on the part of the outcome space where B occurs and ask: what fraction of those outcomes also satisfy A? That fraction is the size of the intersection A ∩ B divided by the size of B, as long as B has positive probability. So the defining formula is P(A|B) = P(A ∩ B) / P(B). This makes intuitive sense: among all outcomes where B occurs, the conditional probability counts only those that also meet A, scaling by how likely B is in the first place. The other expressions don’t align with the standard definition. P(B|A) = P(A ∩ B)/P(A) is a different conditional probability, and dividing it by P(A) would not give P(A|B). The product P(A)P(B) is the joint probability only when A and B are independent, and does not represent P(A|B) in general. Using P(A ∪ B)/P(B) mixes in the union and doesn’t reflect the conditional framework either. Remember to require P(B) > 0, since you can’t condition on an event with zero probability.

Conditional probability tells you the probability of A happening when you already know B has happened. To find that, you focus on the part of the outcome space where B occurs and ask: what fraction of those outcomes also satisfy A? That fraction is the size of the intersection A ∩ B divided by the size of B, as long as B has positive probability. So the defining formula is P(A|B) = P(A ∩ B) / P(B).

This makes intuitive sense: among all outcomes where B occurs, the conditional probability counts only those that also meet A, scaling by how likely B is in the first place. The other expressions don’t align with the standard definition. P(B|A) = P(A ∩ B)/P(A) is a different conditional probability, and dividing it by P(A) would not give P(A|B). The product P(A)P(B) is the joint probability only when A and B are independent, and does not represent P(A|B) in general. Using P(A ∪ B)/P(B) mixes in the union and doesn’t reflect the conditional framework either.

Remember to require P(B) > 0, since you can’t condition on an event with zero probability.